用户: Scavenger/Prismatic Cohomology

Fix a prime number $p$ throughout.

Convention 0.1. We always use homological index convention when talking about chain complexes and related objects such as differential granded algebra, even if we are using superscripts.

We often use the letter “d” (which is short for the word “discrete”) to indicate that we are working in the $1$-categorical or classical context. For example, if $A$ is an ordinary ring, we use ${\mathrm{dMod}}_A$ to mean the $1$-category of discrete (or classical) modules over $A$, and leave the notation ${\mathrm{Mod}}_A$ to denote the $\infty$-category of $A$-module spectra. If $M,N$ are two discrete $A$-modules, then we use $M{\otimes }_A^{d}N$ to denote the classical tensor product, and leave the notation $M{\otimes }_{A}N$ to denote the tensor product of $M$ and $N$ as $A$-module spectra.

As another example, let $A$ be an ordinary ring, $I\subseteq A$ a finitely generated ideal, and $M$ a discrete $A$-module. We can form the derived $I$-completion of $M$ either in the category of discrete $A$-modules (so the result is forced to be discrete) or in the $\infty$-category of $A$-module spectra (so the result is not necessarily discrete). We will refer to the first situation as discrete derived completion and the second situation as the nondiscrete derived completion.

1Preliminaries on Commutative Algebra

Proposition 1.1. Let $A\to B$ be a map of $E_{\infty }$-rings, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal, let $J:=I\cdot {\pi }_{0}B\subseteq {\pi }_{0}B$. Let $M$ be a $B$-module spectrum. Then $M$ is $J$-nilpotent ($J$-local, $J$-complete) if and only if it is $I$-nilpotent ($I$-local, $I$-complete) when being regarded as an $A$-module spectrum.

Proof. The nilpotent case is simple. Let’s consider the local case. Let $f_1,...,f_n$ be a set of generators of $I$. Let $V_i=\mathrm{fib}(A\to A[1/f_i])$ in ${\mathrm{Mod}}_A$, and let $V={⨂}_{1\le i\le n}{V}_i$ in ${\mathrm{Mod}}_A$. Then an $A$-module spectrum $N$ is $I$-local if and only if $V{\otimes }_{A}N=0$. The same construction with $(A,I)$ replaced by $(B,J)$ yields $V^{\prime }=V{\otimes }_{A}B$, and we see that $M{\otimes }_B{V}^{\prime }=M{\otimes }_{A}V$, so the former vanishes precisely when the latter vanishes.

Now we turn to the complete case. An $A$-module spectrum $N$ is $I$-complete if and only if ${\mathrm{Ext}}_A^k(A[1/f_i],N)=0$ for all $k\in \mathbb{Z}$ and $1\le i\le n$. Since ${\mathrm{Ext}}_B^k(B[1/f_i],M)\simeq {\mathrm{Ext}}_B^k(A[1/f_i]{\otimes }_{A}B,M)\simeq {\mathrm{Ext}}_A^k(A[1/f_i],M)$, so the former vanishes precisely when the latter vanishes.

$\square$

1.2. Let $A$ be an ordinary ring and let $I\subseteq A$ be a finitely ideal. Choose a finite set $f_1,...,f_n$ of generators of $I$. Consider the sequence $...\to \mathrm{fib}(\mathrm{Kos}(A;f_1^2,...,f_n^2)\to A/(f_1^2,...,f_n^2))\to \mathrm{fib}(\mathrm{Kos}(A;f_1^1,...,f_n^1)\to A/(f_1^1,...,f_n^1))$ of $A$-module spectra. We assume that this sequence is essentially zero. That is, for every $i\in {\mathbb{Z}}_{\ge 1}$, there exists a $j\in {\mathbb{Z}}_{>i}$ such that the map from the $j$th element to the $i$th element in the above sequence is zero. Let $M$ be an $A$-module spectrum, then tensoring the above sequence with $M$, the resulting sequence is again essentially zero, and consequently its limit is zero. Thus the $I$-completion of $M$ is given by $\lim (...\to M{\otimes }_{A}A/(f_1^2,...,f_n^2)\to M{\otimes }_{A}A/(f_1^1,...,f_n^1))$. If we assume $M$ is flat over $A$ (and in particular $M$ is discrete), then $\lim (...\to M{\otimes }_{A}A/(f_1^2,...,f_n^2)\to M{\otimes }_{A}A/(f_1^1,...,f_n^1))=\lim (...\to M/(f_1^2,...,f_n^2)M\to M/(f_1^1,...,f_n^1)M)$, which is discrete as one can compute, and equals the classical $I$-completion of $M$.

Proposition 1.3. Let $A$ be a $\delta$-ring, $I\subseteq A$ a finitely generated ideal containing $p$. Let $B$ be an ordinary $A$-algebra that is a discrete derived $I$-completion of an ordinary etale $A$-algebra (this condition holds in particular when $B$ is $I$-completely etale). Then $B$ admits a unique $\delta$-structure compatible with the one on $A$.

Proof. We use the characterization of $\delta$-structures in terms of $W_2$. Let $B^{\prime }$ be an etale $A$-algebra, and $B^{\prime }\to B$ a map of $A$-algebras that exhibits $B$ as the discrete derived $I$-completion of $B^{\prime }$. We have a commutative diagram of short exact sequences of discrete $A$-modules

where ${\varphi }_{\ast }{B}^{\prime }$ and ${\varphi }_{\ast }B$ are $B^{\prime }$ and $B$ regarded as discrete $A$-modules through the ring maps $A\stackrel{\varphi }{\to }A\to B^{\prime }$ and $A\stackrel{\varphi }{\to }A\to B$. Since $p\in I$, $\varphi (I)A$ and $I$ have the same radical, thus the discrete derived $I$-completion of ${\varphi }_{\ast }{B}^{\prime }$ is ${\varphi }_{\ast }B$. Since the left and right vertical arrows in the above diagram are discrete derived $I$-completion arrows, so is the middle one.

Consider the commutative diagram

where the left vertical arrow is etale. Since $W_2(B)$ is derived $(p)$-complete, if we let $f:=(0,1)\in W_2(B)$, then $W_2(B)$ is $(f)$-complete. Therefore $W_2(B)\to W_2(B)/(f)$ and hence $W_2(B)\to B$ is a transfinite composition of square zero extensions in $\mathrm{AniRing}$, the $\infty$-category of animated rings, and consequently $W_2(B)\to B$ is right orthogonal to $A\to B^{\prime }$ in $\mathrm{AniRing}$, and there exists a unique map $B^{\prime }\to W_2(B)$ rendering the diagram

commutative. This map then extends to a unique map $B\to W_2(B)$ by the universal property of discrete derived completion, and we are done.

$\square$

From the proof of Proposition 1.3 we obtain the following

Corollary 1.4. Let $A$ be a $\delta$-ring, $I\subseteq A$ a finitely generated ideal containing $p$. Let $B$ be the discrete derived $I$-completion of $A$, equipped with the $\delta$-structure provided by Proposition 1.3. Then for every $\delta$-ring $C$ over $A$ such that $C$ is derived $I$-complete, ${\mathrm{Hom}}_{\delta \text{-Ring}}(B,C)\to {\mathrm{Hom}}_{\delta \text{-Ring}}(A,C)$ is a bijection.

More generally, let $A$ be a $\delta$-ring, $I\subseteq A$ a finitely generated ideal containing $p$. Let $B$ be an ordinary etale $A$-algebra, let $B^{\prime }$ be the discrete derived $I$-completion of the $A$-algebra $B$, equipped with the $\delta$-structure provided by Proposition 1.3. Let $C$ be a $\delta$-ring over $A$ such that $C$ is derived $I$-complete, let $B\to C$ be an $A$-algebra map. Then the $A$-algebra map $B^{\prime }\to C$ provided by the universal property of discrete derived completion is a map of $\delta$-rings.

2Prisms

Definition 2.1. A $\delta$-pair is a pair $(A,I)$ where $A$ is a $\delta$-ring and $I\subseteq A$ is an ideal. A morphism between two $\delta$-pairs $(A,I)$ and $(B,J)$ is a morphism $f:A\to B$ of $\delta$-rings such that $f(I)\subseteq J$.

A prism is a $\delta$-pair $(A,I)$ satisfying the following conditions:

(i) $I$ is a Cartier divisor, which means that the quasicoherent sheaf on $\mathrm{Spec}A$ determined by the discrete $A$-module $I$ is a line bundle.

(ii) $A$ is derived $(p,I)$-complete as a module spectrum over itself.

(iii) $p\in I+\varphi (I)A$.

A morphism of prisms is a morphism of the underlying $\delta$-pairs.

A prism $(A,I)$ is bounded if $A/I$ has bounded $p^{\infty }$-torsion.

Notation 2.2. Let $(A,I)$ be a prism. The ring $\overline{A}:=A/I$ is called the slice of $(A,I)$.

Proposition 2.3 (Rigidity of Prisms). Let $(A,I)\to (B,J)$ be a morphism of prisms. Then the natural map $I{\otimes }_A^{d}B\to J$ is an isomorphism of discrete $B$-modules. In particular $J=IB$.

Proof. See [4] Lemma 5.4.2.

$\square$

2.4. Let $(A,I)$ be a bounded prism. Then the ideal $(p,I)\subseteq A$ satisfies the hypothesis of 1.2, where we use the “set of generators” $\{p^n,I\}$ for some $n\in {\mathbb{Z}}_{\ge 1}$ sufficiently large (here we use the cartier divisor $I$ directly in the Koszul complexes, instead of choosing a set of generators of $I$). Consequently we have the following

Proposition 2.5. Let $(A,I)$ be a bounded prism, let $M$ be a flat discrete $A$-module. Let $\hat{M}$ be the classical $(p,I)$-completion of $M$. Then the map $M\to \hat{M}$ exhibits $\hat{M}$ as the $(p,I)$-completion of $M$ (in $\mathcal{D}(A)$). In particular the $(p,I)$-completion of $M$ is discrete, and $A$ is itself classically $(p,I)$-complete.

Similar things hold if we replace $A$ by $\overline{A}$ and $(p,I)$ by $(p)$.

3The Prismatic Site

Construction 3.1. Let $(A,I)$ be a prism with slice $\overline{A}$. Let $R$ be an $\overline{A}$-algebra. The prismatic site of $R$ relative to $A$, denoted $(R/A{)}_{\mathbf{\Delta }}$, is defined to be the opposite category of the following category, where:

(i) An object consists of a morphism of prisms $(A,I)\to (B,IB)$ together with a morphism of $\overline{A}$-algebras $R\to B/IB$. We will typically notate such an object as $(R\to B/IB\leftarrow B)$ and depict such an object as a diagram below:

(ii) A morphism between two objects $(R\to B/IB\leftarrow B)$ and $(R\to C/IC\leftarrow C)$ will be a morphism of prisms $(B,IC)\to (C,IC)$ compatible with the above diagram.

We will always equip $(R/A{)}_{\mathbf{\Delta }}$ with the indiscrete Grothendieck topology (and in particular every presheaf is a sheaf).

Proposition 3.2. Let $(A,I)$ be a bounded prism and $R$ a $p$-completely smooth $A/I$-algebra. Then there exists a prism $(B,J)$ over $(A,I)$ such that $B/IB\simeq R$ as $A/I$-algebras.

Proof. Use Elkik’s algebraization theorem we choose a smooth $A/I$-algebra $R^{\prime }$ and a map $R^{\prime }\to R$ of $A/I$-algerbas that exhibits $R$ as the nondiscrete derived $(p)$-completion of $R^{\prime }$. Use [8] tag 07M8 to choose a smooth $A$-algebra $S^{\prime }$ such that $S^{\prime }/I{S}^{\prime }\simeq R^{\prime }$ as $A/I$-algebras. Let $S^{\prime }\to S$ be a map of $A$-algebras that exhibits $S$ as the nondiscrete derived $(p,I)$-completion of $S^{\prime }$. By Proposition 2.5 $S$ is discrete and equal to the classical $(p,I)$-completion of $S^{\prime }$. Then $\mathrm{fib}(S^{\prime }\to S)$ is $(p,I)$-local, thus $\mathrm{fib}(S^{\prime }\to S){\otimes }_{A}A/I$ is $p$-local. Since $A$ and $I$ are invertible discrete modules over $A$, tensoring with them does not affect $(p,I)$-nilpotence/locality/completeness, so $S{\otimes }_{A}A/I\simeq \mathrm{cofib}(S{\otimes }_{A}I\to S{\otimes }_{A}A)$ is $(p)$-complete. So $S{\otimes }_{A}A/I$ is the nondiscrete derived $(p)$-completion of $S^{\prime }{\otimes }_{A}A/I\simeq R^{\prime }$, and in particular $S{\otimes }_{A}A/I$ is discrete and isomorphic to $R$ as $A/I$-algebras.

Now using the fact that, for any ordinary ring $B$, a $\delta$-structure on $B$ is equivalent to a section of the projection $W_2(B)\to B$, we can give $S^{\prime }$ a $\delta$-structure compatible with the one of $A$. Using [2] Lemma 2.17 we can give $S$ a $\delta$-structure compatible with the one of $S^{\prime }$. We claim that $(S,IS)$ is a prism. The only nontriviality is to prove that $IS$ is a Cartier divisor. Since $S{\otimes }_{A}A/I$ is discrete, $S{\otimes }_A^{d}I\to S$ is injective, thus $IS\simeq S{\otimes }_A^{d}I$ is a line bundle over $S$. The prism $(S,IS)$ is the prism we are looking for.

$\square$

Construction 3.3 (Prismatic and Hodge-Tate Cohomology). Let $(A,I)$ be a prism with slice $\overline{A}$. Let $R$ be an $\overline{A}$-algebra. We define sheaves of rings ${\mathcal{O}}_{\mathbf{\Delta }}$ and ${\overline{\mathcal{O}}}_{\mathbf{\Delta }}$ on $(R/A{)}_{\mathbf{\Delta }}$ as follows: ${\mathcal{O}}_{\mathbf{\Delta }}((R\to B/IB\leftarrow B))=B$ and ${\overline{\mathcal{O}}}_{\mathbf{\Delta }}((R\to B/IB\leftarrow B))=B/IB$.

The (unique) functor $p:(R/A{)}_{\mathbf{\Delta }}\to \ast$ (where $\ast$ is the category with one object and one morphism) is cocontinuous, thus induces a geometric morphism between $\infty$-topoi (so we can preform pushforwards ($p_{\ast }$) and pullbacks ($p^{\ast }$) along $p$, and $(p^{\ast },p_{\ast })$ is an adjoint pair). In particular we have a functor $p_{\ast }:{\mathrm{Sh}}_{E_{\infty }\text{-Ring}}((R/A{)}_{\mathbf{\Delta }})\to E_{\infty }\text{-Ring}$, where $E_{\infty }\text{-Ring}$ is the $\infty$-category of $E_{\infty }$-rings. Define $E_{\infty }$-rings ${\mathbf{\Delta }}_{R/A}:=p_{\ast }({\mathcal{O}}_{\mathbf{\Delta }})$ and ${\overline{\mathbf{\Delta }}}_{R/A}:=p_{\ast }({\overline{\mathcal{O}}}_{\mathbf{\Delta }})$.

Notice that the pullback functor $p^{\ast }:E_{\infty }\text{-Ring}\to {\mathrm{Sh}}_{E_{\infty }\text{-Ring}}((R/A{)}_{\mathbf{\Delta }})$ is given by composition with $p:(R/A{)}_{\mathbf{\Delta }}\to \ast$, so $p^{\ast }(B)=B$ (the constant sheaf on $B$) for every $E_{\infty }$-ring $B$. Using adjointness we see that we have a natural diagram of $E_{\infty }$-rings:

The natural frobenius endormorphisms $\varphi$ of prisms induces an endormorphism $\varphi :{\mathcal{O}}_{\mathbf{\Delta }}\to {\mathcal{O}}_{\mathbf{\Delta }}$ of the sheaf of rings ${\mathcal{O}}_{\mathbf{\Delta }}$, which is compatible with the frobenius endormorphism of $A$. Applying $p_{\ast }$ (and using adjunction) we see that ${\mathbf{\Delta }}_{R/A}$ admits a natural endormorphism $\varphi$ compatible with the frobenius endormorphism of $A$.

Proposition 3.4. The diagram

is a pushout diagram in $E_{\infty }\text{-Ring}$.

Proof. For any sheaf of $A$-module spectra $M$ on $(R/A{)}_{\mathbf{\Delta }}$ and an $A$-module spectrum $N$, we have a natural map $p_{\ast }M{\otimes }_{A}N\to p_{\ast }(M{\otimes }_A{p}^{\ast }N)$ induced by adjunction, thus functorial in $M$ and $N$. For $N=A$ or $N=I$ this map is easily checked to be an isomorphism, thus this map is an isomorphism for $N=A/I=\mathrm{cofib}(I\to A)$. Taking $M={\mathcal{O}}_{\mathbf{\Delta }}$ we get what we want (notice that ${\overline{\mathcal{O}}}_{\mathbf{\Delta }}={\mathcal{O}}_{\mathbf{\Delta }}{\otimes }_{A}A/I$).

$\square$

Proposition 3.5. ${\mathbf{\Delta }}_{R/A}$ is a $(p,I)$-complete $A$-module spectrum, and consequently ${\overline{\mathbf{\Delta }}}_{R/A}$ is a $(p)$-complete $\overline{A}$-module spectrum.

Proof. For $k\in \mathbb{Z}$ and $x\in (p,I)\subseteq A$, ${\mathrm{Hom}}_{{\mathrm{Mod}}_A}((A[1/x])[k],{\mathbf{\Delta }}_{R/A})\simeq {\mathrm{Hom}}_{{\mathrm{Sh}}_{{\mathrm{Mod}}_A}((R/A{)}_{\mathbf{\Delta }})}((A[1/x])[k],{\mathcal{O}}_{\mathbf{\Delta }})$, so we only need to prove that the latter is contractible. Recall that ${\mathcal{O}}_{\mathbf{\Delta }}:(R/A{)}_{\mathbf{\Delta }}^{\mathrm{op}}\to {\mathrm{Mod}}_A$ is a functor, thus ${\mathrm{Hom}}_{{\mathrm{Sh}}_{{\mathrm{Mod}}_A}((R/A{)}_{\mathbf{\Delta }})}((A[1/x])[k],{\mathcal{O}}_{\mathbf{\Delta }})\simeq {\mathrm{Hom}}_{{\mathrm{Mod}}_A}((A[1/x])[k],\lim ({\mathcal{O}}_{\mathbf{\Delta }}))$ is contractible since $\lim ({\mathcal{O}}_{\mathbf{\Delta }})$ is $(p,I)$-complete.

$\square$

Lemma 3.6. Let $(A,I)$ be a prism. Then the forgetful functor from the category of prisms over $(A,I)$ to the category of $\delta$-pairs over $(A,I)$ admits a left adjoint, which we call the prismatic envelope.

Proof. Let $(B,J)$ be a $\delta$-pair over $(A,I)$. We construct the prismatic envelope for $(B,J)$. For every $\delta$-ring $C$ and discrete $C$-module $M$, there exists a $\delta$-ring $C\{M\}$ over $C$ such that for every $\delta$-ring $X$ over $C$, there exists a bijection from ${\mathrm{Hom}}_{\delta {\text{-Ring}}_{C/}}(C\{M\},X)$ to ${\mathrm{Hom}}_{C\text{-Mod}}(M,X)$, functorial in $X$. Let $F\subseteq I^{-1}{\otimes }_A^{d}B$ denote the subset consisting of elements $x$ such that for every element $i\in I$, $ix\in I{\otimes }_A^d{I}^{-1}{\otimes }_A^{d}B\cong B$ lies in $J$, then $F$ is a sub discrete $B$-module (one can show that $F=I^{-1}{\otimes }_A^{d}J$). Consider the $\delta$-ring $B\{F\}$ together with the universal map $j:F\to B\{F\}$. For every pair of elements $i\in I$ and $f\in F$, consider the element $q(i,f):=l(i)\cdot j(f)-l(i\cdot f)\in B\{F\}$, where $l:B\to B\{F\}$. Let $B^{\prime }$ be the $\delta$-ring obtained form $B\{F\}$ by quotient out elements of the form $q(i,f)$ (that is, $B^{\prime }$ is $B\{F\}$ quotient out the minimum ideal stable under the $\delta$ operation and contains $q(i,f)$, as ordinary rings). Then the map $(B,J)\to (B^{\prime },I{B}^{\prime })$ is a map of $\delta$-pairs. Moreover, if $(C,IC)$ is a prism over $(A,I)$, then the map ${\mathrm{Hom}}_{\delta \text{-Pair}}((B^{\prime },I{B}^{\prime }),(C,IC))\to {\mathrm{Hom}}_{\delta \text{-Pair}}((B,J),(C,IC))$ is a bijection. Summarizing what we’ve done above, we see that we may assume $J=IB$.

Let $B_0:=B$, for every ordinal $i$, let $B_{i+1}^{\prime }$ be the maximum $I$-torsion free quotient of $B_i$ (taken in the context of $\delta$-rings, so $B_{i+1}^{\prime }$ admits a natural $\delta$-structure) (one can easily show that such a maximum $I$-torsion free quotient indeed exists), and let $B_{i+1}$ be the discrete derived $(p,I)$-completion of $B_{i+1}^{\prime }$ (so $B_{i+1}$ naturally admits a $\delta$-structure). Let $B_i={\mathrm{colim}}_{j<i}(B_j)$ (colimit taken in the category of $\delta$-rings) if $i$ is a nonzero limit ordinal. Since the category of derived $(p,I)$-complete discrete $A$-modules is stable under $\kappa$-filtered $1$-colimits, where $\kappa >\omega$ is a regular cardinal, the map $(B,IB)\to (B_{\kappa },I{B}_{\kappa })$ is the prismatic envelope we are looking for.

$\square$

Lemma 3.7. Let $(A,I)$ be a prism, let $R$ be an $\overline{A}$-algebra. Then the category $(R/A{)}_{\mathbf{\Delta }}$ admits finite nonempty products.

Proof. See the proof of [4] Lemma 11.6.4.

$\square$

Proposition 3.8. Let $(A,I)$ be a prism, let $R$ be an $\overline{A}$-algebra. Then the category $(R/A{)}_{\mathbf{\Delta }}$ admits a weakly final object.

Proof. The proof of the corresponding proposition (Proposition 11.6.5) of [4] is wrong. We give a correct proof below.

Let $B_0:=A[R]$ be the polynomial $A$-algebra on the set of free generators $R$, so we have a surjection $B_0\to R$ of $A$-algebras, mapping a free generator $r\in R$ to $r\in R$. Let $B$ be the free $\delta$-ring over $A$ generated by the set $R$, so we have a natural map $B_0\to B$. Let $J=\ker (B_0\to R)$, then one can show that the prismatic envelope of $(B,JB)$ gives a weakly final object.

$\square$

4The Hodge-Tate Comparison Map

4.1 (Totalization). I don’t know how to prove the following facts, but I think they are correct:

Let $A:\Delta \to \mathrm{Ab}$ be a cosimplicial abelian group,where $\mathrm{Ab}$ is the category of abelian groups, then (I think) [3] Problem 4.23 shows that $\lim A$ (limit taken in the derived $\infty$-category $\mathcal{D}(\mathbb{Z})$) is naturally equivalent to the complex associated to the cosimplicial abelian group $A$ whose differentials are given by alternating sums, as in [4] Definition 11.2.2.

The above discussion naturally extends to the case of cosimplicial chain complexes of abelian groups. More precisely, the limit functor $\mathrm{Fun}(\Delta ,\mathrm{Ch}(\mathbb{Z}))\to \mathrm{Fun}(\Delta ,\mathcal{D}(\mathbb{Z}))\stackrel{\lim }{\to }\mathcal{D}(\mathbb{Z})$ is equivalent to the functor $\mathrm{Fun}(\Delta ,\mathrm{Ch}(\mathbb{Z}))\stackrel{f}{\to }\mathrm{Ch}(\mathbb{Z})\to \mathcal{D}(\mathbb{Z})$, where $f$ is the functor first taking a cosimplicial chain complex to the underlying double chain complex, and then to the totalization of this double chain complex. Moreover, I think this equivalence should be compatible with the projection from the limit of a cosimplicial chain complex to the level $0$ chain complex of the cosimplicial chain complex.

Now assume $A:\Delta \to \mathrm{dRing}$ is a cosimplicial commutative ring, where $\mathrm{dRing}$ is the category of ordinary commutative rings. I think the monoidal Dold-Kan correspondence equips the associated complex of abelian groups of $A$ the structure of an $E_{\infty }$-ring (see this nLab page), and this should be the limit of $A$ in $E_{\infty }\text{-Ring}$, the $\infty$-category of $E_{\infty }$-rings. In particular the commutative graded ring structure on the homotopy groups is given by the cup product (see this nLab page), that is, for $a\in A([i])$, $b\in A([j])$, both are cycles of the complex, we have $a\cdot b=(x_{\ast }(a)\in A([i+j]))\cdot (y_{\ast }(b)\in A([i+j]))$, where $x:[i]\to [i+j]$ is the inclusion of the first $i+1$ elements, and $y:[j]\to [i+j]$ is the inclusion of the last $j+1$ elements.

Definition 4.2. Let $A\to B$ be a morphism of ordinary rings. Define the de Rham complex to be the cdga (commutative differential graded algebra) over $A$ given by the chain complex $B\to {\Omega }_{B/A}^1\to {\Omega }_{B/A}^2\to ...$, where ${\Omega }_{B/A}^i={\wedge }_B^i{\Omega }_{B/A}$, whose multiplication is given by the wedge product.

The de Rham complex of $B$ over $A$ will be denoted $({\Omega }_{B/A}^{\bullet },d_{\mathrm{dR}})$, or simply ${\Omega }_{B/A}^{\bullet }$.

Lemma 4.3. Let $A\to B$ be a morphism of ordinary rings. Let $(E^{\bullet },d)$ be a coconnective cdga over $A$. Let $\eta :B\to E^0$ be a map of $A$-algebras such that for each element $x\in B$, $y=d(\eta (x))\in E^{-1}$ satisfies $y^2=0$. Then $\eta$ extends uniquely to a map ${\Omega }_{B/A}^{\bullet }\to E^{\bullet }$ of cdgas over $A$.

Proof. See [4] Lemma 12.2.3.

$\square$

Definition 4.4. Let $A$ be an ordinary ring and $I\subseteq A$ a finitely generated ideal. The subcategory of the category of chain complexes of discrete $A$-modules spanned by the chain complexes of discrete $A$-modules that are levelwise derived $I$-complete is a reflective subcategory, which is compatible with the symmetric monoidal structure in the sense of [5] Definition 2.2.1.6. Consequently the full subcategory of the category of $A$-cdgas spanned by the levelwise derived $I$-complete $A$-cdgas is a reflective subcategory, and the localization functor associated to this reflective subcategory is given by levelwise discrete derived $I$-completion.

The derived $I$-completion of the de Rham complex will be called the completed de Rham complex, denoted ${\hat{\Omega }}_{B/A}^{\bullet }$. It satisfies a similar universal property as that of ordinary de Rham complex.

Computation 4.5. Let $(A,I)$ be a bounded prism. Let $R$ be a $(p)$-completely smooth $\overline{A}$-algebra, so there exists a smooth $\overline{A}$-algebra $R^{\prime }$ and a map $R^{\prime }\to R$ of $\overline{A}$-algebras that exhibits $R$ as the nondiscrete derived $(p)$-completion of $R^{\prime }$. Let $i\in {\mathbb{Z}}_{\ge 1}$. In this case we have a canonical map $f:{\Omega }_{R^{\prime }/\overline{A}}^i{\otimes }_{R^{\prime }}^{d}R\to {\Omega }_{R/\overline{A}}^i$. Since $R^{\prime }$ is flat over $\overline{A}$ and $(A,I)$ is bounded, $R$ is the classical $(p)$-completion of $R^{\prime }$, thus ${\Omega }_{R^{\prime }/\overline{A}}^i{\otimes }_{R^{\prime }}^{d}R$ is the classical $(p)$-completion of ${\Omega }_{R^{\prime }/\overline{A}}^i$ since ${\Omega }_{R^{\prime }/\overline{A}}^i$ is finite projective over $R^{\prime }$. Notice that $f$ becomes an isomorphism after modulo $p^n$, where $n\in {\mathbb{Z}}_{\ge 1}$, so we have a map $g:{\Omega }_{R/\overline{A}}^i\to {\Omega }_{R^{\prime }/\overline{A}}^i{\otimes }_{R^{\prime }}^{d}R$ such that $g\circ f=\mathrm{id}$, so ${\Omega }_{R/\overline{A}}^i=({\Omega }_{R^{\prime }/\overline{A}}^i{\otimes }_{R^{\prime }}^{d}R)\oplus M$, where $M$ is a discrete $R$-module. It is easy to check that $pM=M$, so the discrete derived $(p)$-completion of $M$ vanishes. Therefore the discrete derived $(p)$-completion of ${\Omega }_{R/\overline{A}}^i$ is equal to the classical $(p)$-completion, which is also equal to ${\Omega }_{R^{\prime }/\overline{A}}^i{\otimes }_{R^{\prime }}^{d}R$. Summarizing what we’ve done so far, we see that ${\hat{\Omega }}_{R/\overline{A}}^{\bullet }\simeq {\Omega }_{R^{\prime }/\overline{A}}^{\bullet }{\otimes }_{R^{\prime }}^{d}R$ (the completion is the levelwise classical $(p)$-completion here), and in particular each term of it is finite projective over $R$.

Construction 4.6. Let $(A,I)$ be a prism, let $R$ be an $\overline{A}$-algebra. For $M$ a discrete $\overline{A}$-module and $n\in \mathbb{Z}$, define the Breuil-Kisin twist $M\{n\}:=M{\otimes }_{\overline{A}}^d(I/I^2{)}^{{\otimes }^{d}n}$.

Let $M$ be an $A$-module spectrum. For $n\in \mathbb{Z}$, tensoring the fiber sequence $I^{n+1}/I^{n+2}\to I^n/I^{n+2}\to I^n/I^{n+1}$ in ${\mathrm{Mod}}_A$ with $M$ we obtain a fiber sequence $M{\otimes }_A{I}^{n+1}/I^{n+2}\to M{\otimes }_A{I}^n/I^{n+2}\to M{\otimes }_A{I}^n/I^{n+1}$. Considering the connecting homomorphism we obtain a map ${\beta }^n:{\pi }_{-n}(M{\otimes }_A{I}^n/I^{n+1})\to {\pi }_{-n-1}(M{\otimes }_A{I}^{n+1}/I^{n+2})$ of discrete $A$-modules. These maps gives the ${\pi }_{-n}(M{\otimes }_A{I}^n/I^{n+1})$’s the structure of a chain complex of discrete $A$-modules by [4] Lemma 12.3.2. Moreover, if $M$ can be promoted to an $E_{\infty }$-ring over $A$, then this chain complex has a natural structure of an $A$-cdga (the Lebniz rule can be easily seen by observing that $A/I^2\to A/I$ is a square zero extension of animated rings by the $A/I$-module $I/I^2$).

In particular letting $M={\mathbf{\Delta }}_{R/A}$ we get a cdga structure on ${\pi }_{-n}({\overline{\mathbf{\Delta }}}_{R/A})\{n\}$.

Proposition 4.7. Let $\eta :R\to {\pi }_0({\overline{\mathbf{\Delta }}}_{R/A})$ be the map induced by $\eta :R\to {\overline{\mathbf{\Delta }}}_{R/A}$. For any element $t\in R$, ${\beta }^1(\eta (t))\in {\pi }_{-1}({\overline{\mathbf{\Delta }}}_{R/A})\{1\}$ squares to zero in ${\pi }_{-2}({\overline{\mathbf{\Delta }}}_{R/A})\{2\}$.

A proof of the above proposition, implicitly using the unproved claim 4.1, is given in [4] Lemma 12.3.4 (there are some calculation errors that don’t matter essentially). For the readers feeling unsafe about this unproved claim, there is always a functorial approach, see the comment after Lemma 5.4 of [1] Lecture 5, at the top of page 8.

Using the universal property of completed de Rham complex, we get a map of cdgas ${\hat{\Omega }}_{R/\overline{A}}^{\bullet }\to {\pi }_{-\bullet }({\overline{\mathbf{\Delta }}}_{R/A})\{\bullet \}$ (the completion is the $(p)$-completion here). This map is called the Hodge-Tate comparison map.

Now we can formulate the Hodge-Tate comparison theorem:

Theorem 4.8 (Hodge-Tate Comparison Theorem). Let $(A,I)$ be a bounded prism. Let $R$ be a $(p)$-completely smooth $\overline{A}$-algebra. Then the Hodge-Tate comparison map constructed above is an isomorphism of cdgas.

5Proof of the Hodge-Tate Comparison Theorem

Definition 5.1 (Divided Power Envelope). Let $R$ be an ordinary ring that is flat over $\mathbb{Z}$ (or equivalently $R$ is $n$-torsion free for $n\in {\mathbb{Z}}_{\ne 0}$). The divided power operations ${\gamma }_n:R\to R{\otimes }_{\mathbb{Z}}^d\mathbb{Q}$ are the maps of sets defined by the formula ${\gamma }_n(x)=\frac{x^n}{n!},x\in R,n\in {\mathbb{Z}}_{\ge 0}$.

Let $J$ be an ideal of $R$. The divided power envelope of (R,J) is the subring $D$ of $R{\otimes }_{\mathbb{Z}}^d\mathbb{Q}$ generated by $R$ and ${\gamma }_n(x)$ for all $x\in J$ and $n\in {\mathbb{Z}}_{\ge 0}$. We also denote this ring by $D_J(R)$. If $\{x_i{\}}_{i\in I}$ is a set of generators of the ideal $J$, then $D_J(R)$ is equal to the subring of $R{\otimes }_{\mathbb{Z}}^d\mathbb{Q}$ generated by $R$ and ${\gamma }_n(x_i)$ for all $i\in I$ and $n\in {\mathbb{Z}}_{\ge 0}$.

Aside 5.2 (Cosimplicial Objects). Let $\mathcal{C}$ be an ordinary category. The category $\mathrm{Fun}(\Delta ,\mathcal{C})$ is called the category of cosimplicial objects of $\mathcal{C}$ (here $\Delta$ is the simplex category), and its elements are called cosimplicial objects of $\mathcal{C}$.

If $\mathcal{C}$ admits finite products, then $\mathrm{Fun}(\Delta ,\mathcal{C})$ admits a canonical simplicial enrichment. In this case for $X$ a finite simplicial set and $A$ a cosimplicial object of $\mathcal{C}$, we define a new cosimplicial object of $\mathcal{C}$, denoted $A^X$, by the formula $A^X([n])={\prod }_{X_n}A([n])$, with the obvious transition maps. Now for $A,B$ two cosimplicial objects of $\mathcal{C}$ we define a simplicial set $\mathrm{Hom}(A,B)$ by the formula $\mathrm{Hom}(A,B{)}_n={\mathrm{Hom}}_{\mathrm{Fun}(\Delta ,\mathcal{C})}(A,B^{{\Delta }^n})$. It is easy to check that this gives $\mathrm{Fun}(\Delta ,\mathcal{C})$ a simplicial enrichment. Moreover, if $\mathcal{D}$ is another ordinary category which admits finite products, and $\mathcal{C}\to \mathcal{D}$ is a functor preserving finite products, then the induced functor $\mathrm{Fun}(\Delta ,\mathcal{C})\to \mathrm{Fun}(\Delta ,\mathcal{D})$ can be naturally promoted into a simplicially enriched functor.

Let $\mathcal{C}$ be a category which admits finite products. The simplicial category $\mathrm{Fun}(\Delta ,\mathcal{C})$ has a natural homotopy category, which we denote by $\mathrm{hFun}(\Delta ,\mathcal{C})$, obtained by taking ${\pi }_0$ at each $\mathrm{Hom}$-simplicial set of $\mathrm{Fun}(\Delta ,\mathcal{C})$. Let $A,B$ be two cosimplicial objects of $\mathcal{C}$, and let $f,g:A\to B$ be two morphisms of cosimplicial objects of $\mathcal{C}$. $f$ and $g$ are called homotopic if they define the same morphism in $\mathrm{hFun}(\Delta ,\mathcal{C})$. $f$ is said to be a homotopy equivalence if it defines an equivalence in $\mathrm{hFun}(\Delta ,\mathcal{C})$. Let $\mathcal{D}$ be a category which admits finite products and let $F:\mathcal{C}\to \mathcal{D}$ be a functor preserving finite products, then $F:\mathrm{Fun}(\Delta ,\mathcal{C})\to \mathrm{Fun}(\Delta ,\mathcal{D})$ preserves homotopic morphisms and homotopy equivalences.

Remark 5.3. The notions of homotopic morphisms and homotopy equivalences between cosimplicial objects of a category $\mathcal{C}$ can actually be defined even if the $\mathcal{C}$ is not assumed to have finite products, and these notions are preserved by an arbitrary functor $F:\mathcal{C}\to \mathcal{D}$ between categories, even if $F$ is not assumed to preserve finite products. See [8] tag 0FKJ for details.

Let $\mathrm{dMod}$ be the $1$-category of pairs $(A,M)$ where $A$ is an ordinary ring and $M$ is a discrete $A$-module, so we a natural forgetful functor $\mathrm{dMod}\to \mathrm{dRing}$, where $\mathrm{dRing}$ is the category of ordinary rings, mapping a pair $(A,M)$ to the ring $A$. The induced functor $\mathrm{Fun}(\Delta ,\mathrm{dMod})\to \mathrm{Fun}(\Delta ,\mathrm{dRing})$ can be naturally promoted to a simplicially enriched functor. Let $A$ be a cosimplicial ring, we let ${\mathrm{Mod}}_A:=\mathrm{Fun}(\Delta ,\mathrm{dMod}){\times }_{\mathrm{Fun}(\Delta ,\mathrm{dRing})}\{A\}$, which has a natural simplicial enrichement, and consequently a corresponding theory of homotopic morphisms and homotopy equivalences. Objects of ${\mathrm{Mod}}_A$ will be called cosimplicial $A$-modules. We have a natural tensor product operation in ${\mathrm{Mod}}_A$, which is formed by levelwise tensor product. Let $f,f^{\prime }:M\to M^{\prime }$ be two homotopic morphisms of cosimplicial $A$-modules, let $N$ be a cosimplicial $A$-module, then it is easy to show that $f\otimes N$ and $f^{\prime }\otimes N$ are homotopic.

Let $\mathcal{A}$ be an abelian category. For a cosimplicial object $A$ of $\mathcal{A}$ we have the associated Moore complex $s(A)$ (see [8] tag 019H for definitions), which is a chain complex concerntrated in nonpositive degrees, so in particular $s(A{)}_{-n}=A([n])$ for $n\in {\mathbb{Z}}_{\ge 0}$. A morphism $A\to B$ of cosimplicial objects of $\mathcal{A}$ is called a weak equivalence if $s(A)\to s(B)$ is a quasi-isomorphism. It is easy to check that the functor $s:\mathrm{Fun}(\Delta ,\mathcal{A})\to {\mathrm{Ch}}_{\le 0}(\mathcal{A})$ descends to a functor $s:\mathrm{hFun}(\Delta ,\mathcal{A})\to {\mathrm{hCh}}_{\le 0}(\mathcal{A})$, where ${\mathrm{hCh}}_{\le 0}(\mathcal{A})$ is the homotopy category of nonpositively graded chain complexes (see [8] tag 01A0). In particular, homotopy equivalences between cosimplicial objects of $\mathcal{A}$ are weak equivalences.

Proposition 5.4 (Cech Nerves). Let $f:X\to Y$ be a morphism of a category $\mathcal{C}$ which admits pushouts. Assume that there is a morphism $s:Y\to X$ such that $s\circ f={\mathrm{id}}_X$. Consider the Cech nerve $U$ (or better, opposite Cech nerve, see [8] tag 016N for an explicit construction) of $f$ (this is a cosimplicial object in $\mathcal{C}$). The morphism $U\to U$ which in each degree is the self map of $Y{\coprod }_X...{\coprod }_{X}Y$ given by $f\circ s$ on each factor is homotopic to ${\mathrm{id}}_U$. In particular $U$ is homotopy equivalent to the constant cosimplicial object on $X$.

Proof. See [8] tag 019Z.

$\square$

Lemma 5.5 ([2] Lemma 5.4). Let $k$ be an ordinary commutative ring of characteristic $p$. Let $B$ be a polynomial $k$-algebra. Let $B^{\bullet }$ be the (opposite) Cech nerve of $k\to B$, and let $B^{\bullet ,(1)}$ be its Frobenius twist (given by (levelwise) pushing out along the Frobenius endomorphism of $k$), so we have the relative Frobenius $B^{\bullet ,(1)}\to B^{\bullet }$ as a map of cosimplicial $k$-algebras. Then for any cosimplicial discrete $B^{\bullet ,(1)}$-module $N$, the natural map $N\to N{\otimes }_{B^{\bullet ,(1)}}^d{B}^{\bullet }$ (the tensor product here is given by levelwise tensor product) is a weak equivalence of cosimplicial discrete $k$-modules.

Notation 5.6. Let $A$ be a $\delta$-ring. Let $J\subseteq A$ be an ideal, we let $J_{\delta }$ be the minimum ideal of $A$ containing $J$ and stable under $\delta$. Let $x_1,...,x_n,d\in A$ be elements (where $n\in {\mathbb{Z}}_{\ge 0}$). We use $A\{\frac{x_1,...,x_n}{d}\}$ to denote the $\delta$-ring $A\{t_1,...,t_n\}/(d{t}_1-x_1,...,d{t}_n-x_n{)}_{\delta }$.

Proposition 5.7. Let $m\le n\in {\mathbb{Z}}_{\ge 1}$. Let $B:={\mathbb{Z}}_p\{x_1,...,x_n\}$ be the free $\delta$-ring generated by the free variables $x_1,...,x_n$. Let $J\subseteq B$ be the ideal generated by $x_1,...,x_m$. Then there is a canonical isomorphism of rings $B\{\frac{\varphi (x_i),1\le i\le m}{p}\}\cong D_J(B)$.

Proof. By definition, $B\{\frac{\varphi (x_i),1\le i\le m}{p}\}={\mathbb{Z}}_p\{x_1,...,x_n,t_1,...,t_m\}/(p{t}_1-\varphi (x_1),...,p{t}_m-\varphi (x_m){)}_{\delta }$. By the proof of [4] Lemma 14.3.2, $D_J(B)$ is a sub $\delta$-ring of $B{\otimes }_{\mathbb{Z}}^d\mathbb{Q}\cong {\mathbb{Q}}_p\{x_1,...,x_n\}$. Consider the map of $\delta$-rings $B\{\frac{\varphi (x_i),1\le i\le m}{p}\}\to D_J(B)$, mapping $t_i$ ($1\le i\le m$) to $\frac{\varphi (x_i)}{p}$ and $x_j$ ( $1\le j\le n$) to $x_j$. One can show that this map is an isomorphism (see [1] Lecture 6, Lemma 2.1, [4] Corollary 14.3.3 and [2] Lemma 2.36).

$\square$

Aside 5.8. See this mathoverflow page for discussions around the question of how to get an $E_{\infty }$-ring from a cdga. The most direct reason for this is the fact that $\mathrm{Ch}(\mathbb{Z})\to \mathrm{Ch}(\mathbb{Z})[w^{-1}]$ is lax symmetric monoidal, see [6] Theorem A.7. There is also another explanation for this construction, as in [7] Example 3.3.14:

Let $k$ be an ordinary commutative ring. Then the symmetric monoidal $\infty$-category of completely filtered k-module spectra has a compatible $t$-structure, called the Beilinson $t$-structure, whose heart coincides with the $1$-category of cochain complexes. The compatibility implies that the inclusion of the heart has a lax symmetric monoidal structure, thus a cdga gives rise to a filtered $E_{\infty }$-$k$-algebra. Passing to the underlying $E_{\infty }$-algebra, we get what we want.

Computation 5.9 (Crystalline-de Rham Comparison for Affine Spaces). Let $r,s\in {\mathbb{Z}}_{\ge 0}$. Let $R={\mathbb{F}}_p[x_1,...,x_r]$, $P_0={\mathbb{Z}}_p[x_1,...,x_r]$, $P={\mathbb{Z}}_p[x_1,...,x_r,y_1,...,y_s]$. Let $D$ be the divided power envelope of $(P,(y_1,...,y_s))$. Since $D$ is a subring of $P{\otimes }_{{\mathbb{Z}}_p}^d{\mathbb{Q}}_p$, one can naturally form a cdga $D{\otimes }_P^d{\Omega }_{P/Z_p}^{\bullet }$. Then one can show that the map ${\Omega }_{P_0/Z_p}^{\bullet }\to D{\otimes }_P^d{\Omega }_{P/Z_p}^{\bullet }$ is a quasi-isomorphism of cdgas by reducing to the case where $r=0,s=1$ and then conclude using explicit calculation (notice that the cdgas we are considering are repeated tensor products of cdgas corresponding to the case $(r,s)=(0,1)\text{ or }(1,0)$).

Using the same argument as above one can show that the map ${\Omega }_{P_0/Z_p}^{\bullet }\to D{\otimes }_P^d{\Omega }_{P/Z_p}^{\bullet }$ remains a quasi-isomorphism after levelwise modulo $p$, hence also remains a quasi-isomorphism after levelwise classical $(p)$-completion by derived Nakayama lemma ([8] tag 0G1U) since both ${\Omega }_{P_0/Z_p}^{\bullet }$ and $D{\otimes }_P^d{\Omega }_{P/Z_p}^{\bullet }$ are $p$-torsion free.

Computation 5.10 (Hodge-Tate Comparison Theorem for Affine Spaces). In the context of Construction 4.6, let $(A,I)=({\mathbb{Z}}_p,(p))$, $R={\mathbb{F}}_p[x_1,...,x_n]$ (where $n\in {\mathbb{Z}}_{\ge 1}$). Let $B:={\mathbb{Z}}_p\{x_1,...,x_n\}$ be the free $\delta$-ring generated by the free variables $x_1,...,x_n$, let $B^{\wedge }$ be the classical $(p)$-completion of $B$. Then it is easy to see that $R\cong B^{\wedge }/p{B}^{\wedge }$ and $(B^{\wedge },p{B}^{\wedge })$ is a weakly final object of $(R/A{)}_{\mathbf{\Delta }}$. We now give an explicit formula for the Cech nerve of $(A,I)\to (B^{\wedge },p{B}^{\wedge })$ in $(R/A{)}_{\mathbf{\Delta }}^{\mathrm{op}}$.

Let $B^{\bullet }$ be the Cech nerve of $A\to B$ formed in the category of $\delta$-rings, so $B^i=A\{x_j^{(k)},0\le k\le i,1\le j\le n\}$. One can check that $C^i:=B^i\{\frac{x_j^{(k)}-x_j^{(0)},1\le k\le i,1\le j\le n}{p}\}$ is a free $\delta$-ring over ${\mathbb{Z}}_p$ and in particular $p$-torsion free. Therefore the underlying cosimplicial ring of the Cech nerve of $(A,I)\to (B^{\wedge },p{B}^{\wedge })$ in $(R/A{)}_{\mathbf{\Delta }}^{\mathrm{op}}$ is (isomorphic to) $(C^{\bullet }{)}^{\wedge }$, where $(C^i{)}^{\wedge }$ is the classical $(p)$-completion of $C^i$.

One can show that there is a canonical isomorphism of rings $C^i{\otimes }_{B^i,\varphi }^d{B}^i\cong B^i\{\frac{\varphi (x_j^{(k)}-x_j^{(0)}),1\le k\le i,1\le j\le n}{p}\}=:D^i$, where the first ring is the pushout of $C^i$ along the Frobenius endomorphism of $B^i$. We show that $f:(C^{\bullet }{)}^{\wedge }\to (D^{\bullet }{)}^{\wedge }$ is a weak equivalence of cosimplicial rings (where the superscript $\wedge$ means classical $p$-completion). By virtue of Lemma 5.5, $f$ is a weak equivalence after quotient out $p$. As $(C^{\bullet }{)}^{\wedge }$ and $(D^{\bullet }{)}^{\wedge }$ are levelwise $p$-torsion free ($D^{\bullet }$ is $p$-torsion free since it can be identified with the divided power envelope of a free $\delta$-ring over ${\mathbb{Z}}_p$, by Proposition 5.7. $C^{\bullet }$ is $p$-torsion free since it is a free $\delta$-ring over ${\mathbb{Z}}_p$), we see that their mod $p$ reductions equal their derived mod $p$ reduction (i.e. derived tensoring with ${\otimes }_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z}$), so the totalization (i.e. taking limit of the corresponding cosimplicial objects in the $\infty$-category of $E_{\infty }$-rings or other suitable $\infty$-categories) of $f$ (regarded as a morphism in $\mathcal{D}(\mathbb{Z})$) is a quasi-isomorphism after derived mod $p$ reduction. Since the totalizations of $(C^{\bullet }{)}^{\wedge }$ and $(D^{\bullet }{)}^{\wedge }$ are limits of derived $(p)$-complete discrete modules, they are derived $(p)$-complete when being regarded as objects of $\mathcal{D}(\mathbb{Z})$, so by derived Nakayama lemma ([8] tag 0G1U), the totalization of $f$ is an equivalence in $\mathcal{D}(\mathbb{Z})$, and consequently $f$ is a weak equivalence of cosimplicial rings.

Let $P^{\bullet }$ be the cosimplicial ring whose level $i$ ($i\in {\mathbb{Z}}_{\ge 1}$) ring $P^i$ is ${\mathbb{Z}}_p[x_j^{(k)},0\le k\le i,1\le j\le n]$. Let $Q^{\bullet }$ be the cosimplicial ring whose level $i$ ring $Q^i$ is ${\mathbb{Z}}_p[x_j^{(k,l)},0\le k\le i,1\le j\le n,l\in {\mathbb{Z}}_{\ge 1}]$ (one should think $x_j^{(k,l)}$ as denoting the $l$-fold $\delta$ operation of $x_j^{(k)}$ in $B^i$). Let $J_i\subseteq P^i$ be the ideal generated by $x_j^{(k)}-x_j^{(0)},1\le k\le i,1\le j\le n$. One easily sees by Proposition 5.7 that the cosimplicial ring $D^{\bullet }$ is isomorphic to the cosimplicial ring $D_{J_{\bullet }}(P^{\bullet }){\otimes }_{{\mathbb{Z}}_p}{Q}^{\bullet }$. By [8] tag 023M the map ${\mathbb{Z}}_p\to Q^{\bullet }$ of cosimplicial rings is a weak equivalence, so consequently $D_{J_{\bullet }}(P^{\bullet })\to D_{J_{\bullet }}(P^{\bullet }){\otimes }_{{\mathbb{Z}}_p}{Q}^{\bullet }\cong D^{\bullet }$ is also a weak equivalence of cosimplicial rings (the cosimplicial ring $D_{J_{\bullet }}(P^{\bullet }){\otimes }_{{\mathbb{Z}}_p}{Q}^{\bullet }$ is the diagonal of the bicosimplicial ring (that is, a functor $\Delta \times \Delta \to \mathrm{dRing}$) $D_{J_{\bullet }}(P^{\bullet }){\otimes }_{{\mathbb{Z}}_p}{Q}^{\ast }$, so their limits agree. Notice the limit of the latter bicosimplicial ring when regarded as a bicosimplicial discrete ${\mathbb{Z}}_p$-module is the totalization of the underlying double chain complex). The same argument as above shows that $D_{J_{\bullet }}(P^{\bullet })/p{D}_{J_{\bullet }}(P^{\bullet })\to D^{\bullet }/p{D}^{\bullet }$ is a weak equivalence, so using derived Nakayama lemma ([8] tag 0G1U) we see that $D_{J_{\bullet }}(P^{\bullet }{)}^{\wedge }\to (D^{\bullet }{)}^{\wedge }$ is a weak equivalence, where $\wedge$ here means levelwise classical $(p)$-completion.

Consider the following double chain complex:

which we think of as the underlying double chain complex of a cosimplicial cdga, in which every morphism is a quasi-isomorphism of cdgas by virtue of Computation 5.9. Each row except the first row is quasi-isomorphic to zero by [4] Lemma 14.4.1, so by considering the spectral sequence associated to this double complex we see that the totalization of the underlying cosimplicial $E_{\infty }$-ring of this cosimplicial cdga is equivalent to the totalization of the first row when being regarded as a cosimplicial ring. Similar things hold when we levelwise modulo $p$, so similar things hold when we take levelwise classical $(p)$-completion by derived Nakayama lemma since everything in the above double complex is $p$-torsion free. Summarizing the above discussion we see that the totalization of the cosimplicial ring $D_{J_{\bullet }}(P^{\bullet }{)}^{\wedge }$ in the $\infty$-category of $E_{\infty }$-ring is equivalent to the underlying $E_{\infty }$-ring of the cdga $(D_{J_0}(P^0){\otimes }_{P^0}^d{\Omega }_{P^0/{\mathbb{Z}}_p}^{\bullet }{)}^{\wedge }\cong ({\Omega }_{P^0/{\mathbb{Z}}_p}^{\bullet }{)}^{\wedge }$. Moreover, under the complicated chain of equivalences $(C^{\bullet }{)}^{\wedge }\simeq ({\Omega }_{P^0/{\mathbb{Z}}_p}^{\bullet }{)}^{\wedge }$ obtained by summarizing all the efforts we’ve made above, the map $R\to {\pi }_0((C^{\bullet }{)}^{\wedge }{\otimes }_A\overline{A})$ corresponds to the map $R\to {\pi }_0(({\Omega }_{P^0/{\mathbb{Z}}_p}^{\bullet }{)}^{\wedge }{\otimes }_A\overline{A})$, mapping each free variable $x_i\in R$ ($1\le i\le n$) to the element $x^p\in P^0/p{P}^0=({\Omega }_{P^0/{\mathbb{Z}}_p}^0{)}^{\wedge }{\otimes }_A\overline{A}$. Using the Cartier isomorphism (see [4] Lemma 14.1.6) we see that the Hodge-Tate comparison theorem holds in this case.

Construction 5.11 (Functoriality of Prismatic Cohomology). Let $(A,I)$ be a prism. Let $R$, $S$ be two ordinary $\overline{A}$-algebras and let $R\to S$ be a map of $\overline{A}$-algebras. We have a natural restriction functor $(S/A{)}_{\mathbf{\Delta }}\to (R/A{)}_{\mathbf{\Delta }}$ and consequently a sequence of pullback functors $E_{\infty }\text{-Ring}\to {\mathrm{Sh}}_{E_{\infty }\text{-Ring}}((R/A{)}_{\mathbf{\Delta }})\to {\mathrm{Sh}}_{E_{\infty }\text{-Ring}}((S/A{)}_{\mathbf{\Delta }})$ in which every functor admits a right adjoint by adjoint functor theorem. Using adjointness we see that we have a natural commutative diagram of $E_{\infty }$-rings:

References

[1]	Bhargav Bhatt, “Geometric Aspects of p-adic Hodge Theory: Prismatic Cohomology”, Eilenberg lectures at Columbia University (fall 2018)
[2]	Bhargav Bhatt, Peter Scholze, “Prisms and Prismatic Cohomology”, arXiv:1905.08229v4.
[3]	Ulrich Bunke, “Differential Cohomology”, arXiv:1208.3961v6.
[4]	Kiran S. Kedlaya, “Notes on Prismatic Cohomology”, https://kskedlaya.org/prismatic/frontmatter-1.html
[5]	Jacob Lurie, “Higher Algebra”, Avalible for download at https://www.math.ias.edu/ lurie/papers/HA.pdf
[6]	Thomas Nikolaus, Peter Scholze, “On Topological Cyclic Homology”, Acta Math 221 (2018), no. 2, 203–409.
[7]	Arpon Raksit, “Hochschild Homology and Derived de Rham Complex Revisited”, arXiv:2007.02576.
[8]	The Stacks project authors, “The Stacks project”, https://stacks.math.columbia.edu